Speaking after Manchester City's 2-0 win at Wigan today Mancini admitted he understood the concerns of Given, who said this week he was frustrated by his lack of first-team opportunities.
Mancini said: "I have a lot of respect for Shay and I told him the situation before the finish of the [transfer] market. I told him. 'Shay, you can choose. I hope that you stay here but I respect your decision and if you want to leave you can.'
"He decided to stay here and until January he must stay here.
"I understand his frustration but I need that Shay stays here and puts in 100%. In January if he decides to leave I respect his decision."
Given's successor Joe Hart hardly had a shot to save all afternoon as goals from inspirational captain Carlos Tevez and Yaya Toure helped City ease to victory.
Mancini was full of praise for Tevez and said he believed his recent decision to make the Argentinian captain had breathed new life into his game.
Mancini added: "I thought about it during the summer and I spoke to Carlos about it. He is an important player and he can improve as captain and we can have more of him.
"I'm very happy for him because at this moment he probably isn't at 100% of his form. Maybe his goal can help him and it is fantastic he got it today."
For Wigan boss Roberto Martinez it was another case of defensive blunders costing his side dear with Mohamed Diame and Maynor Figueroa at fault for City's goals.
Martinez said: "We created Manchester City's best chances and that is very disappointing. In any performance the hardest thing should be to concede a goal.
"At the moment from time to time we come out with this softness in our defensive play which really lets ourselves down and puts games out of our reach.
"In defence we showed a lack of intensity. We are giving ourselves a mountain to climb to get points from games and that is what is so disappointing."
Martinez accepted Nigel de Jong may have been fortunate to stay on the pitch after two clattering challenges but he refused to seek any other excuses.
"He was lucky but it was as self-inflicted defeat," Martinez added. "I will never look at anything else other than ourselves when it is so obvious what made the difference."